FERMILAB–Pub–95/068-A RU-95-18 astro-ph/9504081 April 1995 Submitted to Phys. Rev. D
Light photinos as dark matter
arXiv:astro-ph/9504081v1 24 Apr 1995
Glennys R. Farrar∗ Department of Physics and Astronomy Rutgers University, Piscataway, NJ 08855 Edward W. Kolb† NASA/Fermilab Astrophysics Center Fermi National Accelerator Laboratory, Batavia, IL 60510, and Department of Astronomy and Astrophysics, Enrico Fermi Institute The University of Chicago, Chicago, IL 60637 There are good reasons to consider models of low-energy supersymmetry with very light photinos and gluinos. In a wide class of models the lightest Rodd, color-singlet state containing a gluino, the R0 , has a mass in the 1-2 GeV range and the slightly lighter photino, γe , would survive as the relic R-odd
species. For the light photino masses considered here, previous calculations
resulted in an unacceptable photino relic abundance. But we point out that processes other than photino self-annihilation determine the relic abundance when the photino and R0 are close in mass. Including R0 ←→ γe processes,
we find that the photino relic abundance is most sensitive to the R0 -to-γe
mass ratio, and within model uncertainties, a critical density in photinos may be obtained for an R0 -to-γe mass ratio in the range 1.2 to 2.2. We
propose photinos in the mass range of 500 MeV to 1.6 GeV as a dark matter candidate, and discuss a strategy to test the hypothesis. PACS number(s): 98.80.Cq, 14.80.Ly, 11.30.Pb ∗ †
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I. INTRODUCTION
In this paper we study the early-Universe evolution and freeze out of light, long-lived or stable, R-odd states, the photino γe and the gluino ge.1 In the type of models we
consider, the photino should be the relic R-odd particle, even though it may be more massive than the gluino. This is because below the confinement transition the gluino is bound into a color-singlet hadron, the R0 , whose mass (which is in the 1 to 2 GeV range when the gluino is very light [1,2]) is greater than that of the photino. Including previously neglected reactions associated with the gluino (more precisely, associated with the R0 ), we find that light photinos may be cosmologically acceptable; indeed they are an attractive dark-matter candidate. In the minimal susy model, the mass matrix of the charged and neutral susy fermions (gauginos and Higgsinos) are determined by Lagrangian terms involving the Higgs chiral f and H f , and the SU(2) and U(1) gauge superfields, W f a and B, e plus soft superfields, H 1 2
supersymmetry breaking terms. This leads to a neutralino mass matrix in the basis e W f 3, H f0 , H f0 ) of the form (B, 1 2
M1 0 −mZ cos β sin θW mZ sin β sin θW 0 M2 mZ cos β cos θW −mZ sin β cos θW −mZ cos β sin θW mZ cos β cos θW 0 −µ mZ sin β sin θW −mZ sin β cos θW −µ 0
.
(1)
Here mZ is the mass of the Z-boson, θW is the Weinberg angle, µ is the coefficient of a supersymmetric mixing term between the Higgs superfields, and tan β is the ratio of the vacuum expectation values of the two Higgs fields responsible for electroweak symmetry breaking. The susy-breaking masses M1 and M2 are commonly assumed to 1
R-parity is a multiplicative quantum number, exactly conserved in most susy models, under which
ordinary particles have R = +1 while new “superpartners” have R = −1. Throughout this paper we will assume that R-parity is exact so the lightest R-odd particle is stable.
1
be of order mZ or larger, and if the susy model is embedded in a grand unified theory, then 3M1 /M2 = 5α1 /α2 . The terms in the Lagrangian proportional to M1 and M2 arise from dimension-3 susy-breaking operators. However such susy-breaking terms are not without problems. It appears difficult to break susy dynamically in a way that produces dimension-3 terms while avoiding problems associated with the addition of gauge-singlet superfields [3]. In models where susy is broken dynamically or spontaneously in the hidden sector and there are no gauge singlets, all dimension-3 susy-breaking operators in the effective low-energy theory are suppressed by a factor of hΦi/mP l , where hΦi is the vacuum expectation value of some hidden-sector field. Thus, dimension-3 terms effectively do not contribute to the neutralino mass matrix. This would imply that at the tree level the gluino is massless, and the neutralino mass matrix is given by Eq. (1) with vanishing (00) and (11) entries. However, a non-zero gluino mass, as well as non-zero entries in the neutralino mass matrix are generated through radiative corrections such as the top-stop loop, and for the neutralinos, “electroweak” loops involving higgsinos and/or winos and binos. The generation of radiative gaugino masses in the absence of dimension-3 susy breaking was studied by Farrar and Masiero [4].2 From Figs. 4 and 5 of that paper one sees that as M0 , the typical susy-breaking scalar mass, varies between 100 and 400 GeV, the gluino mass ranges from about 700 to about 100 MeV,3 while the photino4 mass ranges 2
See also [5] for general formulae. Earlier studies [6,7] of radiative corrections when tree level gaugino
masses are absent included another dimension-3 operator, the so-called “A term,” and did not consider the electroweak loop contributions to the neutralino mass matrix. They also assumed model-dependent relations between parameters. 3
Actually, larger values of M0 are not considered in order to keep the gluino mass greater than about
100 MeV. Otherwise an unacceptably light pseudoscalar meson would be produced [1]. 4
Upon diagonalization of the mass matrix, the physical neutralino states are a linear combination
e0, W f3 , H e 0 , and H e 0 . When the gaugino submatrix elements are small, the lightest neutralino is of B 1 2
2
> 40 GeV. This estimate for the photino mass should from around 400 to 900 MeV, for µ ∼
be considered as merely indicative of its possible value, since an approximation for the electroweak loop used in Ref. [4] is strictly valid only when µ or M0 are much larger than mW . The other neutralinos are much heavier, and the production rates of the photino
and the next-lightest neutralino in Z 0 decay are consistent with lep bounds [4]. Using the results of Ref. [4], but additionally restricting parameters so that the correct electroweak symmetry breaking is obtained, Farrar [2] found M0 ∼ 150 GeV and estimated the R0 lifetime. This allowed completion of the study of the main phenomenological features of this scenario, which was begun in Ref. [1]. The conclusion is that light gluinos and photinos are quite consistent with present experiments, and result in a number of striking predictions [2]. However models with light gauginos have been widely thought to be disallowed because it has been believed that the relic density of the lightest neutralino, usually referred to as the lsp,5 exceeds cosmological bounds unless R-parity is violated [8–9]. In this paper we point out that previous considerations of the relic abundance have neglected the rather important interplay between the photino and the gluino which can determine the final neutralino abundance if the photino and gluino are both light, as they must be in models without dimension-3 explicit susy-breaking terms. We find that when gluino–photino interactions are included, rather than being a cosmological embarrassment, these very light photinos are an excellent dark matter candidate. In this paper we discuss the decoupling and relic abundance of light photinos, and the sensitivity of the result upon the parameters of the susy models. f 3 and B e 0 that is almost identical to the SU(2) × U(1) composition of the a linear combination of W
photon, and thus is correctly called “photino.” 5
In this scenario, lsp is an ambiguous term: the gluino is lighter than the photino, although the
photino is lighter than the R0 . A more relevant term would be lrocs—lightest R-odd color singlet.
3
For the light masses studied here, freeze-out occurs well after the confinement transition so the physical states must be color singlets. Since ge is not a color singlet, below
the confinement transition the relevant state to consider is the lightest color-singlet state containing a gluino, which is believed to be a gluon-gluino bound state known as the R0 . The other light R-odd states are more massive than these, and decay to the two light ones with lifetimes much faster than the expansion rate at freeze out. The only other possible state of interest is the S 0 , which is the lightest R-odd baryon, consisting of the
color-singlet, flavor-singlet state udsge [1,2,10]. The masses for the states we consider will
be assumed to be in the range [2,4] ge(gluino) :
γe (photino) : R0 (geg) :
S 0 (udsge) :
meg = 100 − 600 MeV
m = 100 MeV − 1400 MeV
M = 1 − 2 GeV MS 0 = 1.5 − 2 GeV.
(2)
Since it is the lightest color-singlet R-odd state, the γe is stable, and R0 decays to a final state consisting of a photino and typically one meson: R0 −→ γe π; γe η, etc. The lifetime is very uncertain, but probably lies in the range 10−4 to 10−10 s, or even longer [2].
The reaction rates that determine freeze out will depend upon the γe and R0 masses,
the cross sections involving the γe and R0 , and possibly the decay width of the R0 as well.
In turn the cross sections and decay width also depend on the masses of the γe , ge and R0 , as well as the masses of the squarks and sleptons. We will denote the squark/slepton
masses by a common mass scale MSe (expected to be of order 100 GeV). Even if the
masses were known and the short-distance physics specified, calculation of the width
and some of the cross sections would be no easy task, because one is dealing with light hadrons. Fortunately, our conclusions are reasonably insensitive to individual masses, lifetimes, and cross sections, but depend crucially upon the R0 -to-γe mass ratio. 4
When we do need an explicit value of the photino mass m or the masses of squarks and sleptons, we will parameterize them by the dimensionless ratios µ8 ≡
m ; 800 MeV
µS ≡
MSe . 100 GeV
(3)
Although there are several undetermined parameters in our calculation, as mentioned above, the most important parameter will be the the ratio of the R0 mass to the γe mass: r≡
M . m
(4)
This is by far the most crucial parameter, with the relic abundance having an exponential dependence upon r. We find that limits to the magnitude of the contribution to < 2.2,6 while r must be larger the present mass density from relic photinos requires r ∼
than about 1.2 if the photino relic density is to be significant. This narrow band of r
encompasses the large uncertainties in lifetimes and cross sections. If the mass ratio is between about 1.6 and 2, then light-mass relic photinos could dominate the Universe and provide the dark matter with Ωeγ ∼ 1.
In the concluding section we explore the proposal that light photinos are the dark
matter, and discuss possibilities for testing the idea. We lay the groundwork for this suggestion in the next section as we develop a new scenario for decoupling and freezeout for the photinos and gluinos. In Section III we consider the cross sections and lifetimes used in Section IV to calculate the reaction rates relevant for the determination of the freeze-out abundance of the photinos (and hence Ωeγ h2 ). In Section V we compare the reaction rates to the expansion rate and estimate when photinos decouple.
6
Or else R-parity must be violated so the photinos decay.
5
II. SCENARIO FOR PHOTINO/GLUINO FREEZE OUT
The standard procedure for the calculation of the present number density of a thermal relic of the early-Universe is to assume that the particle species was once in thermal equilibrium until at some point the rates for self-annihilation and pair-creation processes became much smaller than the expansion rate, and the particle species effectively froze out of equilibrium. After freeze out, its number density decreased only because of the dilution due to the expansion of the Universe. (For a discussion, see Ref. [11].) Since subsequent to freeze out the number of particles in a co-moving volume is constant, it is convenient to express the number density of the particle species in terms of the entropy density, since the entropy in a co-moving volume is also constant for most of the history of the Universe. The number density-to-entropy ratio is usually denoted by Y . If a species of mass m is in equilibrium and non-relativistic, Y is simply given in terms of the mass-to-temperature ratio x ≡ m/T as YEQ (x) = 0.145(g/g∗)x3/2 exp(−x),
(5)
where g is the number of spin degrees of freedom, and g∗ is the total number of relativistic degrees of freedom in the Universe at temperature T = m/x. Well after freeze-out Y (x) is constant, and we will denote this asymptotic value of Y as Y∞ . If self annihilation determines the final abundance of a species, Y∞ can be found by integrating the Boltzmann equation (dot denotes d/dt)
n˙ + 3Hn = −h|v|σA i n2 − n2EQ ,
(6)
where n is the actual number density, nEQ is the equilibrium density, H is the expansion rate of the Universe, and h|v|σA i is the thermal average [12,13] of the annihilation rate. There are no general closed-form solutions to the Boltzmann equation, but there are reliable, well tested approximations for the late-time solution, i.e., Y∞ . Then with 6
knowledge of Y∞ , the contribution to Ωh2 from the species can easily be found. Let us specialize to the survival of photinos assuming self-annihilation determines freeze out. Calculation of the relic abundance involves first calculating the value of x, known as xf , where the abundance starts to depart from the equilibrium abundance. Using standard approximate solutions to the Boltzmann equation [11] gives7 xf = ln(0.0481mP l mσ0 ) − 1.5 ln[ln(0.0481mP l mσ0 )] ,
(7)
where we have used g = 2 and g∗ = 10, and parameterized the non-relativistic annihilation cross section as h|v|σA i = σ0 x−1 . In anticipation of the results of the next section, 3 4 we use σ0 = 2 × 10−11 µ28 µ−4 S mb, and we find xf ≃ 12.3 + ln(µ8 /µS ). The value of xf
determines Y∞ : Y∞ =
2.4x2f 4 ≃ 7.4 × 10−7 µ−3 8 µS . mP l mσ0
(8)
Once Y∞ is known, the present photino energy density can be easily calculated: ρeγ = mneγ = 0.8 µ8GeV · Y∞ 2970 cm−3 . When this result is divided by the critical
density, ρC = 1.054h2 × 10−5 GeV cm−3 , the fraction of the critical density contributed 4 by the photino is Ωeγ h2 = 2.25 × 108 µ8 Y∞ . For Y∞ in Eq. (8), Ωeγ h2 = 167µ−2 8 µS .
The age of the Universe restricts Ωeγ h2 to be less than one, so for µS = 1, the photino
must be more massive than about 10 GeV if its relic abundance is determined by selfannihilation.
But in this paper we point out that for models in which both the photino and the gluino are light, freeze-out is not determined by photino self annihilation, but by γe –R0
interconversion. The basic point is that since the R0 has strong interactions, it will stay in equilibrium longer than the photino, even though it is more massive. As long 7
Freeze-out aficionados will notice that we use the formulae appropriate for p-wave annihilation
because Fermi statistics requires the initial identical Majorana fermions to be in an L = 1 state [8].
7
as γe ↔ R0 interconversion occurs at a rate larger than H, then through its interactions
with the R0 the photino will be able to maintain its equilibrium abundance even after
self-annihilation has frozen out.8 Before we demonstrate that this scenario naturally occurs for the types of photino and R0 masses expected, we must determine the cross sections and decay width of the reactions involving the photino and the gluino.
III. CROSS SECTIONS AND DECAY WIDTH
In this section we characterize the cross sections and decay width required for the determination of the relic photino abundance, and also discuss the uncertainties. We should emphasize that all cross sections are calculated in the non-relativistic (N.R.) limit, and by h· · ·i we imply that the quantity is to be evaluated as a thermal average [12,13]. In the N.R. limit a temperature dependence to the cross sections enters if the annihilation proceeds through a p-wave, as required if the initial state consists of identical fermions [8]. For p-wave annihilation, at low energy the cross section is proportional to v 2 , where |v| is the relative velocity of the initial particles. The thermal average reduces 8
Actually, interconversion can also play an important role in determining the relic abundance of
heavier photinos. When the photino is more massive and freeze-out occurs above the confinement phase transition, the analysis is similar to the one here; in fact simpler because perturbation theory can be used to compute the relevant rates involving gluinos and photinos. Since the qualitative relation between interconversion and self-annihilation rates is independent of whether the gluino is free or confined in an R0 , one can get a crude idea of the required gluino-photino mass ratio, r, just by using the analysis in this paper and scaling the results to the value of µ8 of interest. We concentrate on the light gaugino scenario because it is attractive in its own right, and also because it naturally produces r in the right ballpark[2]. In a conventional susy-breaking scheme fine-tuning is generally necessary to give r the right value for the interconversion mechanism to play an important role.
8
to replacing v 2 by 6T /m, where m is the mass of the particle in the initial state. We now consider in turn the cross sections and width for the individual reactions discussed in the previous section.
A. Self-annihilations and co-annihilation The first type of reactions we will consider are those which change the number of R-odd particles. R0 R0 → X: We will refer to this process as R0 self-annihilation. At the constituent level the relevant reactions are ge + ge → g +g and ge + ge → q+ q¯, which are unsuppressed by
any powers of MSe, and should be typical of strong interaction cross sections. In the N.R.
limit, we expect the R0 R0 annihilation cross section to be comparable to the p¯p cross
section, but with an extra factor of v 2 , accounting for the fact that there are identical fermions in the initial state, so annihilation must proceed through a p-wave.9 There is some energy dependence to the p¯p cross section, but it is sufficient to consider h|v|σR0 R0 i to be a constant, approximately given by h|v|σR0 R0 i ≃ 100v 2 mb = 600 x−1 r −1 mb,
(9)
where we have used for the relative velocity v 2 = 6T /M = 6/(rx), with x ≡ m/T . We should note that the thermal average of the cross section might be even larger if there are resonances near threshold. In any case, this cross section should be much larger than any cross section involving the photino, and will ensure that the R0 remains in equilibrium longer than the γe , greatly simplifying our considerations. 9
γe γe → X: In photino self-annihilation at low energies the final state X is a lepton-
In general the result is not so simple. For instance, in addition to the term proportional to v 2 ,
the cross section also involves a term proportional to the square of the masses of the initial and final particles.
9
antilepton pair, or a quark-antiquark pair which appears as light mesons. The process involves the t-channel exchange of a virtual squark or slepton between the photinos, producing the final-state fermion-antifermion pair. In the low-energy limit the mass √ MSe of the squark/slepton is much greater than s, and the photino-photino-fermion-
antifermion operator appears in the low-energy theory with a coefficient proportional to e2i /MSe2 , with ei the charge of the final-state fermion.10 Also, as there are two identical
fermions in the initial state, the annihilation proceeds as a p-wave, which introduces
a factor of v 2 in the low-energy cross section [8]. The resultant low-energy photino self-annihilation cross section is [8,9,14,15]: 2 h|v|σeγeγ i = 8παEM
X i
qi4
h i m2 v 2 2 −4 −11 −1 µ µ mb, ≃ 2.0 × 10 x 8 S MSe4 3
(10)
where we have used for the relative velocity v 2 = 6/x with x ≡ m/T , and qi is the magnitude of the charge of a final-state fermion in units of the electron charge. For the light photinos we consider, summing over e, µ, and three colors of u, d, and s quarks leads to
P
4 i qi
= 8/3.
γe R0 → X: This is an example of a phenomenon known as co-annihilation, whereby
the particle of interest (in our case the photino) disappears by annihilating with another particle (here, the R0 ). Of course co-annihilation also leads to a net decrease in R-odd particles. In all processes involving the photino–R0 interaction, the leading tree-level shortdistance operator containing ge and γe is λe†g λeγ qi† qi + h.c., with coefficient eqi gS /MSe2 . For
three light quarks,
P
2 i qi
= 2. Thus we can estimate the cross section for γe R0 → X in
terms of the γe self annihilation cross section: 10
h|v|σeγR0 i ≃
αS 4 2 M 3 h|v|σeγeγ i, αEM 3 8/3 m v 2
(11)
The electric charge e and the strong charge gS are to be evaluated at a scale of order MSe, so in numerical estimates we use αEM = 1/128 and αS = 0.117.
10
where the ratio of α’s arises because the short-distance operator for co-annihilation is proportional to e2i gS2 rather than e4i , the second factor is the color factor coming from the gluino coupling, and the third factor comes from the ratio of
P
2 P 4 i qi / i qi
for the
participating fermions. We have replaced m2 appearing in Eq. (10) by mM, although the actual dependence on m and M may be more complicated. Finally, the annihilation is s-wave so there is no v 2 /3 suppression as in photino self-annihilation. Although the short-distance physics is perturbative, the initial gluino appears in a light hadron, and there are complications in the momentum fraction of the R0 carried by the gluino and other non-perturbative effects. For our purposes it will be sufficient to account for the uncertainty by including in the cross section an unknown coefficient A, leading to a final expression h
i
h|v|σeγR0 i ≃ 1.5 × 10−10 r µ28 µ−4 S A mb.
(12)
It is reassuring that if one estimates h|v|σeγR0 i starting from h|v|σR0 R0 i a similar result is
obtained. We find that co-annihilation will not be important unless A is larger than 102 or so, which we believe is unlikely.
B. γe –R0 interconversion
In what we call interconversion processes, there is an R-odd particle in the initial as well as the final state. Although the reactions do not of themselves change the number of R-odd particles, they keep the photinos in equilibrium with the R0 s, which in turn are kept in equilibrium through their self annihilations. R0 → γe π: R0 decay can occur via, e.g., the gluino inside the initial R0 turning into an
antiquark and a virtual squark, followed by squark decay into a photino and a quark. In the low-energy limit the quark–antiquark–gluino–photino vertex can be described by the same type of four-Fermi interaction as in co-annihilation. One expects on dimensional 11
grounds a decay width Γ0 ∝ αEM αS M 5 /MSe4 . The lifetime of a free gluino to decay to
a photino and massless quark-antiquark pair was computed in Ref. [16]. However this
does not provide a very useful estimate when the gluino mass is less than the photino mass. The lifetime for R0 decay was studied in Ref. [2]. In an attempt to account for the effects of gluino-gluon interactions in the R0 , necessary for even a crude estimate of the R0 lifetime, the following picture was developed: The R0 is viewed as a state with a massless gluon carrying momentum fraction x, and a gluino carrying momentum √ fraction (1 − x),11 having therefore an effective mass M 1 − x. The gluon structure function F (x) gives the probability in an interval x to x + dx of finding a gluon, and the corresponding effective mass for the gluino. One then obtains the R0 decay width (neglecting the mass of final state hadrons): Γ(M, r) = Γ0 (M, 0)
Z
0
1−r −2
√ dx (1 − x)5/2 F (x) f (1/r 1 − x),
(13)
where Γ0 (M, 0) is the rate for a gluino of mass M to decay to a massless photino, and f (y) = [(1 − y 2)(1 + 2y − 7y 2 + 20y 3 − 7y 4 + 2y 5 + y 6 ) + 24y 3 (1 − y + y 2 ) log y] contains the phase space suppression which is important when the photino becomes massive in comparison to the gluino. Modeling K ± decay in a similar manner underestimates the lifetime by a factor of 2 to 4. This is in surprisingly good agreement; however caution should be exercised when extending the model to R0 decay, because kaon decay is much less sensitive to the phase-space suppression from the final state masses than the present case, since the range of interest will turn out to be r ∼ 1.2 − 2.2. For r in this range, taking F (x) ∼ 6x(1 − x) following the discussion in Ref. [2] leads to an approximate 11
Of course there should be no confusion with the fact that in the discussion of the R0 lifetime we use
x to denote the gluon momentum fraction whereas throughout the rest of the paper x denotes m/T .
12
behavior ΓR0 →eγ π = 2.0 × 10−14 F (r) GeV [µ58 µ−4 S B] ,
(14)
where F (r) = r 5 (1 − r −1 )6 , and the factor B reflects the overall uncertainty. We believe
